Need help proving a congruence statement

*Sorry if this isn't the correct section. I'm not too sure about the naming of topics and my course is simply called "Discrete Mathematics" and I got an infraction last time for posting in this thread that's why I'm saying.*

I put [*1] because my teacher said that "that is not always true". Is he right? If so, can someone please elaborate? For [*2], how do we know that? For [*3], I don't see the relevance of that step in the grand scheme of things.